masterarbeit/presentation/presentation.md

---
title:       Evaluation of the Performance of Randomized FFD Control Grids
subtitle:    Master Thesis
author:      Stefan Dresselhaus
affiliation: Graphics & Geometry Group
...

# Introduction

- Many modern industrial design processes require advanced optimization methods
  due to increased complexity
- Examples are
    - physical domains
        - aerodynamics (i.e. drag)
        - fluid dynamics (i.e. throughput of liquid)
    - NP-hard problems 
        - layouting of circuit boards
        - stacking of 3D--objects

-----

# Motivation

- Evolutionary algorithms cope especially well with these problem domains
    ![Example of the use of evolutionary algorithms in automotive design](../arbeit/img/Evo_overview.png)
- But formulation can be tricky

-----

# Motivation

- Problems tend to be very complex
    - i.e. a surface with $n$ vertices has $3\cdot n$ Degrees of Freedom (DoF).
- Need for a small-dimensional representation that manipulates the
  high-dimensional problem-space.
- We concentrate on smooth deformations ($C^3$-continuous)
- But what representation is good?

-----

# What representation is good?

- In biological evolution this measure is called *evolvability*.
    - no consensus on definition
    - meaning varies from context to context
    - measurable?
- Measure depends on representation as well.

-----

# RBF and FFD

- Andreas Richter uses Radial Basis Functions (RBF) to smoothly deform meshes

![Example of RBF--based deformation and FFD targeting the same mesh.](../arbeit/img/deformations.png)

-----

# RBF and FFD

- My master thesis transferred his idea to Freeform-Deformation (FFD)
    - same setup
    - same measurements
    - same results?

![Example of RBF--based deformation and FFD targeting the same mesh.](../arbeit/img/deformations.png)

-----

# Outline

- **What is FFD?**
- What is evolutionary optimization?
- How to measure evolvability?
- Scenarios
- Results

-----

# What is FFD?

- Create a function $s : [0,1[^d \mapsto \mathbb{R}^d$ that is parametrized
    by some special control--points $p_i$ with coefficient functions $a_i(u)$:
    $$
    s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
    $$
- All points inside the convex hull of $\vec{p_i}$ accessed by the right $u \in [0,1[^d$.

![Example of a parametrization of a line with corresponding deformation to generate a deformed objet](../arbeit/img/B-Splines.png)

-----

# Definition B-Splines

- The coefficient functions $a_i(u)$ in $s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}$ are different for each control-point
- Given a degree $d$ and position $\tau_i$ for the $i$th control-point $p_i$ we
  define
  \begin{equation}
  N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases}
  \end{equation}
  and
  \begin{equation} \label{eqn:ffd1d2}
  N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u)
  \end{equation}
- The derivatives of these coefficients are also easy to compute:
  $$\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)$$

# Properties of B-Splines

- Coefficients vanish after $d$ differentiations
- Coefficients are continuous with respect to $u$
- A change in prototypes only deforms the mapping locally  
  (between $p_i$ to $p_{i+d+1}$)

![Example of Basis-Functions for degree $2$. [Brunet, 2010]<br /> Note, that Brunet starts his index at $-d$ opposed to our definition, where we start at $0$.](../arbeit/img/unity.png)

# Definition FFD

- FFD is a space-deformation resulting based on the underlying B-Splines
- Coefficients of space-mapping $s(u) = \sum_j a_j(u) p_j$ for an initial vertex
  $v_i$ are constant
- Set $u_{i,j}~:=~N_{j,d,\tau}$ for each $v_i$ and $p_j$ to get the projection:
    $$
    v_i = \sum_j u_{i,j} \cdot p_j = \vec{u}_i^{T} \vec{p}
    $$
    or written with matrices:
    $$
    \vec{v} = \vec{U} \vec{p}
    $$
- $\vec{U}$ is called **deformation matrix**

# Implementation of FFD

- As we deal with 3D-Models we have to extend the introduced 1D-version
- We get one parameter for each dimension: $u,v,w$ instead of $u$
- Task: Find correct $u,v,w$ for each vertex in our model
    - We used a gradient-descent (via the gauss-newton algorithm)

# Implementation of FFD

- Given $n,m,o$ control-points in $x,y,z$--direction each Point inside the
  convex hull is defined by
    $$V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.$$
- Given a target vertex $\vec{p}^*$ and an initial guess $\vec{p}=V(u,v,w)$
  we define the error--function for the gradient--descent as:
  $$Err(u,v,w,\vec{p}^{*}) = \vec{p}^{*} - V(u,v,w)$$

# Implementation of FFD

- Derivation is straightforward
    $$
    \scriptsize
    \begin{array}{rl}
        \displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \\
      = & \displaystyle - \sum_i \sum_j \sum_k N'_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x
    \end{array}
    $$
  yielding a Jacobian:

$$
\scriptsize
J(Err(u,v,w)) = 
\left(
\begin{array}{ccc}
\frac{\partial Err_x}{\partial u} & \frac{\partial Err_x}{\partial v} & \frac{\partial Err_x}{\partial w} \\
\frac{\partial Err_y}{\partial u} & \frac{\partial Err_y}{\partial v} & \frac{\partial Err_y}{\partial w} \\
\frac{\partial Err_z}{\partial u} & \frac{\partial Err_z}{\partial v} & \frac{\partial Err_z}{\partial w}
\end{array}
\right)
$$

# Implementation of FFD

- Armed with this we iterate the formula
  $$J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)$$
  using Cramer's rule for inverting the small Jacobian.
- Usually terminates after $3$ to $5$ iteration with an $\epsilon := \vec{p^*} -
  V(u,v,w) < 10^{-4}$
- self-intersecting grids can invalidate the results
  - no problem, as these get not generated and contradict some properties we
    want (like locality)

-----

# Outline

- What is FFD?
- **What is evolutionary optimization?**
- How to measure evolvability?
- Scenarios
- Results

-----

# What is evolutionary optimization?

## 1 1

~~~
$t := 0$;
initialize $P(0) := \{\vec{a}_1(0),\dots,\vec{a}_\mu(0)\} \in I^\mu$;
evaluate $F(0) : \{\Phi(x) | x \in P(0)\}$;
while($c(F(t)) \neq$ true) {
    recombine: $P’(t) := r(P(t))$;
    mutate: $P''(t) := m(P’(t))$;
    evaluate $F(t) : \{\Phi(x) | x \in P''(t)\}$
    select: $P(t + 1) := s(P''(t) \cup Q,\Phi)$;
    $t := t + 1$;
}
~~~

~~~
$t$: Iteration-step
$I$: Set of possible Individuals
$P$: Population of Individuals
$F$: Fitness of Individuals
$Q$: Either set of parents or $\emptyset$

$r(..) : I^\mu \mapsto I^\lambda$
$m(..) : I^\lambda \mapsto I^\lambda$
$s(..) : I^{\lambda + \mu} \mapsto I^\mu$
~~~

- Algorithm to model simple inheritance
- Consists of three main steps
    - recombination
    - mutation
    - selection
- An "individual" in our case is the displacement of control-points

# Evolutional loop

- **Recombination** generates $\lambda$ new individuals based on the characteristics of the $\mu$ parents.
    - This makes sure that the next guess is close to the old guess.
- **Mutation** introduces new effects that cannot be produced by mere recombination of the
  parents.
    - Typically these are minor defects to individual members of the population
      i.e. through added noise
- **Selection** selects $\mu$ individuals from the children (and optionally the
    parents) using a *fitness--function* $\Phi$.
    - Fitness could mean low error, good improvement, etc.
    - Fitness not solely determines who survives, there are many possibilities


# Outline

- What is FFD?
- What is evolutionary optimization?
- **How to measure evolvability?**
- Scenarios
- Results

# How to measure evolvability?

- Different (conflicting) optimization targets
    - convergence speed?
    - convergence quality?
- As $\vec{v} = \vec{U}\vec{p}$ is linear, we can also look at $\Delta \vec{v} = \vec{U}\,
    \Delta \vec{p}$
    - We only change $\Delta \vec{p}$, so evolvability should only use $\vec{U}$
      for predictions

# Evolvability criteria

- **Variability**
    - roughly: "How many actual Degrees of Freedom exist?"
    - Defined by
        $$\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n} \in [0..1]$$
    - in FFD this is $1/\#\textrm{CP}$ for the number of control-points used for
      parametrization

# Evolvability criteria

- **Regularity**
    - roughly: "How numerically stable is the optimization?"
    - Defined by
        $$\mathrm{regularity}(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}} \in [0..1]$$
        with $\sigma_{min/max}$ being the least/greatest right singular value.
    - high, when $\|\vec{Up}\| \propto \|\vec{p}\|$

# Evolvability criteria

- **Improvement Potential**
    - roughly: "How good can the best fit become?"
    - Defined by
        $$\mathrm{potential}(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F$$
        with a unit-normed guessed gradient $\vec{G}$

# Outline

- What is FFD?
- What is evolutionary optimization?
- How to measure evolvability?
- **Scenarios**
- Results

# Scenarios

- 2 Testing Scenarios
- 1-dimensional fit
    - $xy$-plane to $xyz$-model, where only the $z$-coordinate changes
    - can be solved analytically with known global optimum
- 3-dimensional fit
    - fit a parametrized sphere into a face
    - cannot be solved analytically
    - number of vertices differ between models

# 1D-Scenario

![Left: A regular $7 \times 4$--grid<br />Right: The same grid after a
random distortion to generate a testcase.](../arbeit/img/example1d_grid.png)

![The target--shape for our 1--dimensional optimization--scenario including a wireframe--overlay of the vertices.](../arbeit/img/1dtarget.png){width=70%}

# 3D-Scenarios

![\newline Left: The sphere we start from with 10 807 vertices<br />Right: The face we want to deform the sphere into with 12 024 vertices.](../arbeit/img/3dtarget.png)

# Outline

- What is FFD?
- What is evolutionary optimization?
- How to measure evolvability?
- Scenarios
- **Results**

# Variability 1D

- Should measure Degrees of Freedom and thus quality

![The squared error for the various grids we examined.<br /> Note that $7 \times 4$ and $4 \times 7$ have the same number of control--points.](../arbeit/img/evolution1d/variability_boxplot.png)

- $5 \times 5$, $7 \times 7$ and $10 \times 10$ have *very strong* correlation ($-r_S = 0.94, p = 0$) between the *variability* and the evolutionary error.

# Variability 3D

- Should measure Degrees of Freedom and thus quality

![The fitting error for the various grids we examined.<br />Note that the number of control--points is a product of the resolution, so $X \times 4 \times 4$ and $4 \times 4 \times X$ have the same number of control--points.](../arbeit/img/evolution3d/variability_boxplot.png)

- $4 \times 4 \times 4$, $5 \times 5 \times 5$ and $6 \times 6 \times 6$ have *very strong* correlation ($-r_S = 0.91, p = 0$) between the *variability* and the evolutionary error.

# Varying Variability

## 1 1

![A high resolution ($10 \times 10$) of control--points over a circle. Yellow/green points contribute to the parametrization, red points don't.<br />An Example--point (blue) is solely determined by the position of the green control--points.](../arbeit/img/enoughCP.png)

![Histogram of ranks of various $10 \times 10 \times 10$ grids with $1000$ control--points each showing in this case how many control--points are actually used in the calculations.](../arbeit/img/evolution3d/variability2_boxplot.png)

# Regularity 1D

- Should measure convergence speed

![Left: *Improvement potential* against number of iterations until convergence<br />Right: *Regularity* against number of iterations until convergence<br />Coloured by their grid--resolution, both with a linear fit over the whole
dataset.](../arbeit/img/evolution1d/55_to_1010_steps.png){width=70%}

- Not in our scenarios - maybe due to the fact that a better solution simply
  takes longer to converge, thus dominating.

# Regularity 3D

- Should measure convergence speed

![Plots of *regularity* against number of iterations for various scenarios together
with a linear fit to indicate trends.](../arbeit/img/evolution3d/regularity_montage.png){width=70%}

- Only *very weak* correlation
- Point that contributes the worst dominates regularity by lowering the least
  right singular value towards 0.

# Improvement Potential in 1D

- Should measure expected quality given a gradient

![*Improvement potential* plotted against the error yielded by the evolutionary optimization for different grid--resolutions](../arbeit/img/evolution1d/55_to_1010_improvement-vs-evo-error.png){width=70%}

- *very strong* correlation of $- r_S = 1.0, p = 0$.
- Even with a distorted gradient

# Improvement Potential in 3D

- Should measure expected quality given a gradient

![Plots of *improvement potential* against error given by our *fitness--function* after convergence together with a linear fit of each of the plotted data to indicate trends.](../arbeit/img/evolution3d/improvement_montage.png){width=70%}

- *weak* to *moderate* correlation within each group.


# Summary

- *Variability* and *Improvement Potential* are good measurements in our cases
- *Regularity* does not work well because of small singular right values
    - But optimizing for regularity *could* still lead to a better grid-setup
      (not shown, but likely)
    - Effect can be dominated by other factors (i.e. better solutions just take
      longer)

# Outlook / Further research

- Only focused on FFD, but will DM-FFD perform better?
    - for RBF the indirect manipulation also performed worse than the direct one
- Do grids with high regularity indeed perform better?

# Thank you

Any questions?